Ordinary fractions are used for counting the parts of a subject. For example, the amount of juice in the glasses can be described by fractions.

In order to write a fraction, you need to know how many parts the whole is divided into and how many of such parts are taken. You know that “half” is a fraction of “one second”, “a third” is a fraction “one third”, “quarter” is a fraction “one fourth”, written as follows:

\( \frac{1}{2} ,\frac{1}{3} , \frac{1}{4}\)

Here, the numbers 2, 3 and 4, stand under the fractional line of the fractions, indicating how many parts the whole is divided into. They “mark” the feature of the division of the whole into parts and they are therefore called denominators. Number 1, which stands above the fractional line of each fraction, shows the number of the taken parts of the whole and therefore is called a numerator.

**Example 1.** Four friends came to James’ birthday. The cake was cut into 8 equal parts. What part of the cake did James eat with his friends, if everyone got one slice each?

**Solution 1.** To answer the task, you need to make a fraction, that is, find out which number is the denominator of the fraction, and which is the numerator. The cake is cut into 8 parts, so the number 8 is the denominator of the fraction. James and his four friends ate five pieces of cake in total, so the number 5 is the numerator of the fraction. Therefore, the children ate \( \frac{5}{8}\) of the cake.

**Example 2.** Let’s practice. Take a look at the next cake and answer what portion of the cake is already eaten.

**Solution 2.** The cake is \(\frac{1}{4}\) eaten.

**\(\bf \underline{Proper\ and\ improper\ fractions}\)
**Look at the next fractions \( \frac{21}{2}, \frac{15}{3}, \frac{78}{4}\) , where the numerator is bigger than the denominator. A fraction where the numerator is less than a denominator is called a

**Proper fraction**, and the other is an improper one. Proper fractions are always less than 1, and improper fractions are bigger than 1 or equal to 1.

**Example 3.** Compare the numerator and the denominator of the fraction\( \frac{15}{23}\) . Determine the type fraction it is.

**Solution 3.** \( 15< 23\frac{15}{23}\) . is a proper fraction.

**Example 4**** . **Using the above diagram as your guide, what fraction of the rectangle below is painted?

**Solution 4.** The number of painted rectangles = 2. The total number of rectangles is 8 . Therefore, the correct answer is \( \frac{2}{8}\) .

**\(\bf\underline{Comparison\ of\ fractions}\)
**When we speak about two fractions where the denominators are the same, the one with the larger numerator is bigger, and the one with the smaller numerator is smaller.

**Example 5**** . **Compare the fractions: \( \frac{100}{158}\) and \(\frac{99}{158}\)

**Solution 5.** These fractions have the same denominators, so by comparing their numerators, where \(100 > 99 \), then \( \frac{100}{158} > \frac{99}{158}\) .

\(\bf\underline{Mixed\ number}\)

- A mixed number is equal to the sum of its whole and fractional part; the fractional part of the mixed number is always the proper fraction.
- The action by which the improper fraction is converted into a mixed number (or a natural number) is called the allocation of the whole part of the improper fraction.
- Rule of allocation of the whole part from the improper fraction.

For the allocation of the whole from the improper fraction, you need to do the following:

1) The numerator of this fraction should be divided by the denominator;

2) Write the quotient as the whole part of the desired mixed number;

3) In the denominator of the fractional part, write the denominator of this fraction;

4) In the numerator of the fractional part, write the remainder of the division.

**Example 6**** .** Allocate the whole part from the fraction \( \frac{32}{5}\).

**Solution 6.**Divide the numerator of this fraction by its denominator. Quotient is equal to 6, and the remainder is equal to 2. Therefore, the whole part of a mixed number is equal to 6, and the numerator of its fractional part is equal to 2. So, we can write: . \(\frac{32}{5} =6 \frac{2}{5}\)

**Example 7**** .** Allocate the whole part from the fraction \( \frac{26}{7} \).

**Solution 7.**.\( 26/7 = 3 \) So, quotient is equal to 3, and the remainder is equal to\( 26 – 7 * 3 = 26 – 21 = 5\) . So, we can write: \( \frac{26}{7} = 3 \frac{5}{7}\).

**Example 8**** .** Transform the mixed numbers \( 8\frac{5}{6}, 3 \frac{2}{7}\) into an improper fraction.

**Solution 8.**\( 8\frac{5}{6} = \frac{8 \cdot 7+5}{6} = \frac{48 + 5}{6} = \frac{53}{6}\) ,

\( 3 \frac{2}{7} = \frac{3 \cdot 7 +2}{7} = \frac{21+2}{7} = \frac{23}{7}\) .

**Example 9**** . **Find \( \frac{3}{4} \) from \( 24\) .

**Solution 9.**\( 24: 4 = 6\) , then \( 6 \cdot 3 = 18\) . So, 18 is \( \frac{3}{4}\) from \( 24\) .

**Example 10**** .** The tourists travelled 24 kilometers. On the first day, they walked \( \frac{3}{8}\) of the whole route, and on the second day the tourists passed \( \frac{2}{3}\) from what they walked on the first day. They travelled the rest of the way on the third day. How many kilometers did the tourists walked in the first two days?

**Solution 10**** .**1) \( 24 : 8 \cdot 3 = 9\) (km) for the first day;

2) \( 9 : 3 \cdot 2 = 6\)(km) for the second day;

3) \( 9 + 6 = 15\) (km) for the first two days.

\(\it Answer: the\ tourists\ traveled\ 15\ km\ in\ the\ first\ two\ days\)

**Adding (subtracting) fractions with the same denominators.
**To find the sum (difference) of two fractions with the same denominators, you need to:

- Write a common denominator in the denominator of the sum (difference);
- Add (subtract)numerators and the write the result in the numerator of the sum (difference).

** Example 11**** . **Find .\( 1) \frac{2}{4} – \frac{1}{4}\) , \( 2) \frac{12}{5} + \frac{2}{5}\) .

**Solution 11.** . \( 1) \frac{2}{4} – \frac{1}{4} = \frac{2 – 1}{4} = \frac{1}{4}\) , \( 2) \frac{12}{5} – \frac{2}{5} = \frac{12 + 2}{5} = \frac{14}{5}\) .

**Adding (subtracting) fractions with different denominators.
**To add or subtract fractions with different denominators, you need to create the same denominators in both fractions.

**Example 12**** . **Find \( 1) \frac{2}{4} – \frac{3}{5}\) , \( 2) \frac{12}{7} + \frac{2}{5}\).

**Solution 12. ** \( 1) \frac{2}{4} – \frac{3}{5} = \frac{2 \cdot 5}{4 \cdot 5} – \frac{3 \cdot 4}{5 \cdot 4} = \frac{10}{20} – \frac{12}{20} = \frac{-2}{20}\) .

\( 2) \frac{12}{7} + \frac{2}{5} = \frac{12 \cdot 5}{7 \cdot 5} + \frac{2 \cdot 7}{5 \cdot 7} = \frac{60}{35} + \frac{14}{35} = \frac{74}{35}\).